Spherical fractional and hypersingular integrals of variable order in generalized Hölder spaces with variable characteristic. (English) Zbl 1220.46018

Samko, Natasha; Vakulov, Boris

Math. Nachr. 284, No. 2-3, 355-369 (2011).

Summary: We consider nonstandard generalized Hölder spaces of functions \(f\) on the unit sphere \(\mathbb S^{n-1}\) in \(\mathbb R^n\), whose local continuity modulus \(\Omega (f, x, h)\) at a point \(x \in \mathbb S^{n-1}\) has a dominant \(\omega (x, h)\) which may vary from point to point. We establish theorems on the mapping properties of spherical potential operators of variable order \(\alpha (x)\), from such a variable generalized Hölder space to another one with a “better” dominant \(\omega _{\alpha }(x, h) = h^{\mathfrak R\alpha (x)}\omega (x, h)\), and similar mapping properties of spherical hypersingular integrals of variable order \(\alpha (x)\) from such a space into the space with “worse” dominant \(\omega _{-\alpha }(x, h) = h^{-\mathfrak R\alpha (x)}\omega (x, h)\). We admit variable complex valued orders \(\alpha (x)\) which may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Hölder spaces with the weight \(\alpha (x)\).

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
47G40	Potential operators
42B15	Multipliers for harmonic analysis in several variables
26A33	Fractional derivatives and integrals

Keywords:

spherical potentials; spherical hypersingular integrals; Matuszewska-Orlicz indices of almost monotonic functions; continuity modulus; generalized Hölder spaces; variable characteristics

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